Comparing and Ordering Whole Numbers

Hello. In the previous lesson, you learned that a digit’s position determines its value: for example, the 5 in $7,503$ means $500$, while the 3 means $3$. That place-value understanding is exactly what lets us decide which whole number is larger.

In this lesson, you will compare whole numbers using $=$, $<$, and $>$, then use those comparisons to put several numbers in ascending or descending order. By the end, you will have a reliable method that works for two-digit numbers and much larger whole numbers.

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The three comparison symbols

A comparison statement tells how two numbers are related.

| Symbol | Read it as | Meaning | Example |
|---|---|---|
| $=$ | is equal to | both numbers have the same value | $46=46$ |
| $<$ | is less than | the number on the left is smaller | $46<64$ |
| $>$ | is greater than | the number on the left is larger | $64>46$ |

The symbols $<$ and $>$ have a wide, open side and a pointed side:

• The open side faces the greater number.
• The point points toward the smaller number.

For example, in

$$18<72$$

the opening faces 72, the greater number. Read the full statement from left to right: “18 is less than 72.”

The same comparison can be stated in reverse:

$$72>18$$

This says, “72 is greater than 18.” Both statements describe the same relationship.

The equals sign is different. It does not say that one number is larger; it says the values match exactly:

$$305=305$$

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Compare the numbers before choosing a symbol

A useful habit is to separate the task into two parts:

1. Decide which number is greater, smaller, or equal using place value.
2. Then choose the symbol that makes a true statement when read from left to right.

Do not begin by guessing which way the symbol should point. First understand the numbers.

This worksheet gives a visual reminder: the open side of each comparison sign faces the larger number, while the equal sign is used when both values match.

Consider:

$$99\square 56$$

Both numbers have two digits, so compare the digits in the greatest place first: the tens place.

• $99$ has 9 tens, or $90$.
• $56$ has 5 tens, or $50$.

Since $90$ is greater than $50$, $99$ is greater than $56$:

$$99>56$$

Now consider a pair with the same tens digit:

$$82\square 83$$

Both have 8 tens, so compare the ones digits. Since $2$ is less than $3$,

$$82<83$$

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The place-value comparison method

For whole numbers written in their usual form, use this method every time.

1. Compare the number of digits. A whole number with more digits is greater.
2. If the numbers have the same number of digits, compare the digits in the leftmost place.
3. If those digits are equal, move one place to the right.
4. Continue until you find the first place where the digits differ.
5. The number with the greater digit in that place is the greater whole number.
6. If every corresponding digit is the same, use $=$.

Why start at the left? The leftmost digit has the greatest place value, so it has the most influence on the number’s size.

For instance, compare:

$$4,218\square 3,999$$

The first number has 4 thousands; the second has 3 thousands. Because $4,000$ is greater than $3,000$, there is no need to compare the remaining digits:

$$4,218>3,999$$

Even though $999$ is a large amount after the 3 in $3,999$, it cannot make up for having one fewer thousand.

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Worked comparisons

Different numbers of digits

Compare:

$$128\square 75$$

The number $128$ has three digits and $75$ has two digits. Any three-digit whole number is greater than any two-digit whole number.

$$128>75$$

You can also view $75$ as $075$: it has 0 hundreds, while $128$ has 1 hundred. Since $1$ hundred is greater than $0$ hundreds, $128$ is larger.

Same leading digit, different later digit

Compare:

$$856\square 842$$

Write the place values mentally:

Place	$856$	$842$
Hundreds	8 hundreds	8 hundreds
Tens	5 tens	4 tens
Ones	6 ones	2 ones

The hundreds digits match, so move to tens. Five tens is greater than four tens. Therefore,

$$856>842$$

The ones digits do not need to be checked. Once a greater place differs, it settles the comparison.

Several matching digits

Compare:

$$134,297\square 137,499$$

Place	First number	Second number	Result
Hundred thousands	1	1	equal
Ten thousands	3	3	equal
Thousands	4	7	second is greater

The first different digits are 4 and 7 in the thousands place. Since $7,000$ is greater than $4,000$,

$$134,297<137,499$$

Notice that you do not compare $297$ with $499$ first. The thousands place has greater value than the hundreds, tens, and ones places together.

Equal numbers

Compare:

$$62,407\square 62,407$$

Every digit in the same place matches, so the numbers have exactly the same value:

$$62,407=62,407$$

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Ordering more than two numbers

Ordering means arranging numbers by size.

• Ascending order goes from smallest to greatest.
• Descending order goes from greatest to smallest.

To order a list, compare the numbers using the same place-value method. Find the smallest or greatest one first, then continue with the remaining numbers.

Suppose the numbers are:

$$4,205,4,025,4,250,3,999$$

For ascending order, begin with the smallest:

• $3,999$ has 3 thousands, while the others have 4 thousands, so it is smallest.
• Of the three numbers beginning with 4 thousands, $4,025$ has 0 hundreds, so it comes next.
• $4,205$ and $4,250$ both have 2 hundreds. Compare tens: $0$ tens is less than $5$ tens.

So the ascending order is:

$$3,999<4,025<4,205<4,250$$

For descending order, list the same values from greatest to smallest:

$$4,250>4,205>4,025>3,999$$

The numbers have not changed; only the order has changed.

A practical check is to read your completed list aloud. In ascending order, each number should be greater than the number before it. In descending order, each number should be less than the number before it.

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Avoiding common mistakes

Comparing from the wrong side

In $4,201$ and $4,199$, it would be a mistake to look at the ones digits first and decide that 9 makes $4,199$ greater. Start on the left:

• Both have 4 thousands.
• The hundreds digits differ: 2 hundreds is greater than 1 hundred.

Therefore,

$$4,201>4,199$$

Forgetting that zero holds a place

Compare:

$$6,080\square 6,800$$

The thousands digits are equal. At the hundreds place, $6,080$ has 0 hundreds, while $6,800$ has 8 hundreds. Therefore,

$$6,080<6,800$$

The zero is important: it tells us there are no hundreds in $6,080$.

Reversing the symbol after finding the answer

Suppose you determine that $42$ is greater than $24$. Both of these are true:

$$42>24$$ $$24<42$$

But this is false:

$$42<24$$

After writing a symbol, read the complete statement from left to right. If the sentence is not true, reverse the symbol.

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Key takeaways

Comparing whole numbers is an application of place value.

• Use $=$ when values are the same, $<$ when the left number is smaller, and $>$ when the left number is greater.
• The open side of $<$ or $>$ faces the greater number.
• A whole number with more digits is greater than one with fewer digits.
• When digit counts match, compare from the leftmost, greatest place value.
• Stop at the first place where the digits differ.
• Ascending order lists numbers from smallest to greatest; descending order lists them from greatest to smallest.

Next, you will use place value again to add whole numbers with a standard written method, lining up ones, tens, hundreds, and larger places carefully.